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WMA12/01 Oct 2022, Question 3: Sigma Notation

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 7 marks

The question, in short

A sequence is defined by aₙ = cos²(nπ/3). Find exact values of early terms, then use them to evaluate a sum from n = 1 to 50 of n + cos²(nπ/3), showing method.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (ai) · 1 mark

Find the exact value of a₁, where aₙ = cos²(nπ/3).

  1. 1a₁ = cos²(π/3)
  2. 2cos(π/3) = 1/2, so a₁ = (1/2)² = 1/4

Answer1/4

Part (aii) · 1 mark

Find the exact value of a₂, where aₙ = cos²(nπ/3).

  1. 1a₂ = cos²(2π/3)
  2. 2cos(2π/3) = −1/2, so a₂ = (−1/2)² = 1/4

Answer1/4

Part (aiii) · 1 mark

Find the exact value of a₃, where aₙ = cos²(nπ/3).

  1. 1a₃ = cos²(π)
  2. 2cos π = −1, so a₃ = (−1)² = 1

Answer1

Part (b) · 4 marks

Using the pattern of cos²(nπ/3), find the exact value of the sum from n = 1 to 50 of (n + cos²(nπ/3)), making your method clear.

  1. 1Split the sum: ∑n + ∑cos²(nπ/3)
  2. 2∑ n from 1 to 50 = (50/2)(1 + 50) = 1275
  3. 3The cos² terms repeat 1/4, 1/4, 1 every 3 terms
  4. 450 terms = 16 full cycles + 2 extra terms: 16 × (1/4 + 1/4 + 1) + 1/4 + 1/4
  5. 5= 16 × 3/2 + 1/2 = 24.5 = 49/2
  6. 6Total = 1275 + 49/2 = 2599/2

Answer2599/2 (= 1299.5)

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Also searched as: WMA12 October 2022 · WMA12/01 · wma12 oct22 · IAS Maths Pure Mathematics 2 October 2022 · question 3.
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